Conformal mapping of periodic bladed boundary

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I am new to conformal mapping and I am trying to find the mapping of the vertical boundary with tilted cuts to a horizontal boundary with perpendicular cuts (See the image). Any suggestion is appreciated.

This image shows how the mapping should work

I tried $w=z^\theta$, it works for the blade at z=0 but does not do what I want for other blades. See this figure.

See the figure of map $w=z^\theta$ for periodic baldes